Lifting $G$-Valued Galois Representations when $\ell \neq p$

Jeremy Booher, Sean Cotner, Shiang Tang

Published 2022-11-07Version 1

Let $\ell$ and $p$ be distinct primes, $F$ an $\ell$-adic field with absolute Galois group $\Gamma_F$, and $k$ a finite field of characteristic $p$. For a split reductive group $G$, we investigate lifting continuous $\overline{\rho} : \Gamma_F \to G(k)$ to characteristic zero. We construct lifts of $\overline{\rho}$ and use the lift to construct a liftable deformation condition for $\overline{\rho}$ provided $p$ is large enough. This generalizes the minimally ramified deformation condition previously studied for classical groups. Doing so involves constructing ``decomposition types'' generalizing the notion of isotypic decomposition of a $\textrm{GL}_n$-valued representation. This requires introducing and studying weakly reductive group schemes: smooth groups schemes with reductive identity component and a finite \'{e}tale component group whose order is invertible on the base. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $G_2$-valued representations.